Problem set 17: definite integrals and applications.
(1) Use integration to calculate the area of a circle.
(2) Determine b > 0 such that the area below 1/x and contained in the set {(x, y) : 0 ≤ y, 1 ≤ x ≤ b} equals 2.
(3) Find a curve y = y(x) such that its length on (0, 1) is infinite.
(4) Find the area of the region enclosed by the parabola y = 3 − x2 and y = −x.
(5) Find the area of the region that is bounded from above by y =√
x and below by the x-axis and the line y = x − 6.
(6) Assume we have a square-based pyramid with side length 2s and height h. Deduce the formula for the volume.
(7) The region between the graph of f (x) = 2 + x sin x and the x-axis over the integral [−2, 2] is revolved about the x-axis to generate a solid. Find the volume of the solid.
(8) Find the area between the x-axis and the graph of f (x) = |cos(x)| over [0, π].
(9) Compute the area between the x-axis and e−x given on the positive real numbers.
(10) Find a Riemann integrable function f not identically to 0 such that Z n+2
n+1
f (x) dx = 1 2
Z n+1 n
f (x) dx for all n ∈ N.
(11) The region in the first quadrant enclosed by the y-axis, the line through x = 2π, and the graphs of y = sin(x) and y = 12sin(x) is revolved around the x-axis.
What is the volume of the generated solid?
(12) Let us look at y = 1 − ax2 defined in the first quadrant for some 0 < a. We have the region bounded by y from above and the x-axis from below. Determine a such that the volume of the solid generated by rotating around the x-axis is the same as rotating around the y-axis.
(13) Let us rotate the disk (x − R − r)2 + y2 ≤ r2 around the y-axis. What is the volume of the generated torus?
(14) Find the length of the curve y = 5x − 4|x| from x = −2 to x = 2.
(15) Find the length of the curve y = x2 from x = −2 to x = 2.
(16) Assume a car is driving with speed sin(πt) for t ∈ [0, 1]. What is its average speed?
(17) Suppose a bird is flying with speed cos(πt) for t ∈ [0, 2]. What is the average speed?
(18) The exponential distribution has probability density function f (x) =
(0, if x < 0 ce−cx, otherwise, where c is a positive constant. We need R∞
−∞f (x) dx = 1. What is c?
(19) Show that R∞
−∞e−x2/2dx is a finite number.
(20) The mean of a random variable X with probability density function f is E(X) =
Z ∞
−∞
xf (x) dx provided the integral converges.
1
2
Let
f (x) = ( 1
x2, if x ≥ 1 0, otherwise.
Show that f is a probability density function and that the distribution has no mean.
